ECON2228 Notes 10 Christopher F Baum Boston College Economics 2014–2015 cfb (BC Econ) ECON2228 Notes 10 2014–2015 1 / 48
Serial correlation and heteroskedasticity in time series regressions Chapter 12: Serial correlation and heteroskedasticity in time series regressions What will happen if we violate the assumption that the errors are not serially correlated , or autocorrelated ? We demonstrated that the OLS estimators are unbiased, even in the presence of autocorrelated errors, as long as the explanatory variables are strictly exogenous. This is analogous to our results in the case of heteroskedasticity, where the presence of heteroskedasticity alone does not cause bias nor inconsistency in the OLS point estimates. However, following that parallel argument, we will be concerned with the properties of our interval estimates and hypothesis tests in the presence of autocorrelation. cfb (BC Econ) ECON2228 Notes 10 2014–2015 2 / 48
Serial correlation and heteroskedasticity in time series regressions OLS is no longer BLUE in the presence of serial correlation, and the OLS standard errors and test statistics are no longer valid, even asymptotically. Consider a first-order Markov error process: u t = ρ u t − 1 + e t , | ρ | < 1 (1) where the e t are uncorrelated random variables with mean zero and constant variance. cfb (BC Econ) ECON2228 Notes 10 2014–2015 3 / 48
Serial correlation and heteroskedasticity in time series regressions What will be the variance of the OLS slope estimator in a simple y on x regression model? For simplicity let us center the x series so that ¯ x = 0 . Then the OLS estimator will be: � T t = 1 x t u t b 1 = β 1 + SST x where SST x is the sum of squares of the x series. cfb (BC Econ) ECON2228 Notes 10 2014–2015 4 / 48
Serial correlation and heteroskedasticity in time series regressions In computing the variance of b 1 , conditional on x , we must account for the serial correlation in the u process: � T � 1 � Var ( b 1 ) = Var x t u t SST 2 x t = 1 � � T � i = 1 x 2 1 t Var ( u t )+ = 2 � T − 1 � T − 1 SST 2 � � j = 1 x t x t − j E u t u t − j x t = 1 � T − 1 T − 1 σ 2 σ 2 � � � ρ j x t x t − j = + 2 SST 2 SST x x t = 1 j = 1 where σ 2 = Var ( u t ) and we have used the fact that � � � � = ρ j σ 2 in the derivation. = Cov E u t u t − j u t u t − j cfb (BC Econ) ECON2228 Notes 10 2014–2015 5 / 48
Serial correlation and heteroskedasticity in time series regressions Notice that the first term in this expression is merely the OLS variance of b 1 in the absence of serial correlation. When will the second term be nonzero? When ρ is nonzero, and the x process itself is autocorrelated, this double summation will have a nonzero value. As nothing prevents the explanatory variables from exhibiting autocorrelation (and in fact many explanatory variables take on similar values through time) the only way in which this second term will vanish is if ρ is zero, and u is not serially correlated. In the presence of serial correlation, the second term will cause the standard OLS variances of our regression parameters to be biased and inconsistent. cfb (BC Econ) ECON2228 Notes 10 2014–2015 6 / 48
Serial correlation and heteroskedasticity in time series regressions In most applications, when serial correlation arises, ρ is positive, so that successive errors are positively correlated. In that case, the second term will be positive as well. Recall that this expression is the true variance of the regression parameter; OLS will only consider the first term. In that case OLS will seriously underestimate the variance of the parameter, and the t − statistic will be much too high. If on the other hand ρ is negative, so that successive errors result from an “overshooting” process, then we may not be able to determine the sign of the second term, since odd terms will be negative and even terms will be positive. Surely, though, it will not be zero. Thus the consequence of serial correlation in the errors, particularly if the autocorrelation is positive, will render the standard t − and F − statistics useless. cfb (BC Econ) ECON2228 Notes 10 2014–2015 7 / 48
Serial correlation in the presence of lagged dependent variables Serial correlation in the presence of lagged dependent variables A case of particular interest, even in the context of simple y on x regression, is that where the explanatory variable is a lagged dependent variable. Suppose that the conditional expectation of y t is linear in its past value: E ( y t | y t − 1 ) = β 0 + β 1 y t − 1 . We can always add an error term to this relation, and write it as y t = β 0 + β 1 y t − 1 + u t (2) cfb (BC Econ) ECON2228 Notes 10 2014–2015 8 / 48
Serial correlation in the presence of lagged dependent variables Let us first assume that the error is “well behaved,” i.e. E ( u t | y t − 1 ) = 0 , so that there is no correlation between the current error and the lagged value of the dependent variable. In this setup the explanatory variable cannot be strictly exogenous, since there is a contemporaneous correlation between y t and u t by construction. In evaluating the consistency of OLS in this context we are concerned with the correlation between the error and y t − 1 , not the correlation with y t , y t − 2 , and so on. In this case, OLS would still yield unbiased and consistent point estimates, with biased standard errors, as we derived above, even if the u process was serially correlated. cfb (BC Econ) ECON2228 Notes 10 2014–2015 9 / 48
Serial correlation in the presence of lagged dependent variables But it is often claimed that the joint presence of a lagged dependent variable and autocorrelated errors, OLS will be inconsistent. This arises, as it happens, from the assumption that the u process in (2) follows a particular autoregressive process, such as the first-order Markov process in (1). If this is the case, then we do have a problem of inconsistency, but it is arising from a different source: the misspecification of the dynamics of the model. cfb (BC Econ) ECON2228 Notes 10 2014–2015 10 / 48
Serial correlation in the presence of lagged dependent variables If we combine (2) with (1), we really have an AR ( 2 ) model for y t , since we can lag (2) one period and substitute it into (1) to rewrite the model as: y t = β 0 + β 1 y t − 1 + ρ ( y t − 1 − β 0 − β 1 y t − 2 ) + e t = β 0 ( 1 − ρ ) + ( β 1 + ρ ) y t − 1 − ρβ 1 y t − 2 + e t = α 0 + α 1 y t − 1 + α 2 y t − 2 + e t (3) so that the conditional expectation of y t properly depends on two lags of y , not merely one. Thus the estimation of (2) via OLS is indeed inconsistent, but the reason for that inconsistency is that y is correctly modelled as AR ( 2 ) . cfb (BC Econ) ECON2228 Notes 10 2014–2015 11 / 48
Serial correlation in the presence of lagged dependent variables The AR ( 1 ) model is seen to be a dynamic misspecification of (3). As is always the case, the omission of relevant explanatory variables will cause bias and inconsistency in OLS estimates, especially if the excluded variables are correlated with the included variables. In this case, that correlation will almost surely be meaningful. To arrive at consistent point estimates of this model, we merely need add y t − 2 to the estimated equation. That does not deal with the inconsistent interval estimates, which will require a different strategy. cfb (BC Econ) ECON2228 Notes 10 2014–2015 12 / 48
Testing for first-order serial correlation Testing for first-order serial correlation As the presence of serial correlation invalidates our standard hypothesis tests and interval estimates, we should be concerned about testing for it. First let us consider testing for serial correlation in the k − variable regression model with strictly exogenous regressors, which rules out, among other things, lagged dependent variables. cfb (BC Econ) ECON2228 Notes 10 2014–2015 13 / 48
Testing for first-order serial correlation The simplest structure which we might posit for serially correlated errors is AR ( 1 ) , the first order Markov process, as given in (1). Let us assume that e t is uncorrelated with the entire past history of the u process, and that e t is homoskedastic. The null hypothesis is H 0 : ρ = 0 in the context of (1). cfb (BC Econ) ECON2228 Notes 10 2014–2015 14 / 48
Testing for first-order serial correlation If we could observe the u process, we could test this hypothesis by estimating (1) directly. Under the maintained assumptions, we can replace the unobservable u t with the OLS residual v t . Thus a regression of the OLS residuals on their own lagged values, v t = κ + ρ v t − 1 + ǫ t , t = 2 , ... T (4) will yield a t − test. That regression can be run with or without an intercept, and the robust option may be used to guard against violations of the homoskedasticity assumption. It is only an asymptotic test, though, and may not have much power in small samples. cfb (BC Econ) ECON2228 Notes 10 2014–2015 15 / 48
Testing for first-order serial correlation A very common strategy in considering the possibility of AR ( 1 ) errors is the Durbin–Watson test, which is also based on the OLS residuals: � T t = 2 ( v t − v t − 1 ) 2 DW = (5) � T t = 1 v 2 t cfb (BC Econ) ECON2228 Notes 10 2014–2015 16 / 48
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